A Semi-Analytical Finite Element Approach in Machine Design of Axisymmetric Structures

Abstract

In the last decades, finite element (FE) method has proven to be an efficient tool for the numerical analysis of twoor three-dimensional structures of whatever complexity, in mechanical, thermal or other physical problems. It is widely recognised that computational cost (as time and computer capability) increases greatly with structure complexity, being larger with three-dimensional analyses than with two-dimensional ones. It is therefore desirable to devise simplified approaches that may provide a reduction in overall computational effort. An example of considerable importance is the study of bodies of revolution (axisymmetric structures) under axisymmetric loading, where a threedimensional problem is solved by a two-dimensional analysis. Examples are vessels under internal pressure, rotating disks, foundation piles (Zienkiewicz & Taylor, 2000). Apart from axisymmetric problems solved by a plane model, a full three-dimensional analysis is needed, in principle, whenever the structure is axisymmetric but the load is not. In such situations, often encountered in many engineering applications, it is desirable to search for simplified approaches, which may still replace (and thus avoid the computation effort needed by) the use of full three-dimensional simulations. A particular sub-class of problems is encountered when the load applied to axisymmetric structures is exactly antisymmetric; an example is a shaft under a torsion load, which, as it will be shown, can be solved by a plane FE approach, which greatly simplifies the analysis. Another example is represented by semi-analytical methods, which have been developed more than fifty years ago for FE analysis of axisymmetric structures loaded nonaxisymmetrically (Wilson, 1965). Such methods use a Fourier series expansion to reduce a three-dimensional problem to a two-dimensional harmonic model and to compute the solution as superposition of results of every harmonic component analysis. At present, this approach is still not well established and it is rarely used in mechanical design. Even commercial FE codes including harmonic elements have found limited application, due to practical difficulties related to Fourier series conversion of external loads. Only few applications of semi-analytical methods to engineering practical cases have been reported in literature, see (Genta & Tonoli, 1996; Lai & Booker, 1991; Kim et al., 1994; Pedersen & Laursen, 1982; Taiebait & Carter, 2001; Thomas et al., 1983; Zienkiewicz & Taylor, 2000). The goal of this work is twofold. First, it aims to provide a theoretical background on the use of semi-analytical FE approach in numerical analysis of axisymmetric structures loaded nonaxialsymmetrically. In particular, two original results are developed: a plane axi-